Equimultiplicity of topologically equisingular families of parametrized surfaces in ${\mathbb{C}}^3$

We provide a positive answer to Zariski’s conjecture for families of singular surfaces in ${\mathbb{C}}^3,$ under the condition that the family has a smooth normalisation. As a corollary of the result, we obtain a surprising characterization of the Whitney equisingularity of one parameter families of $\mathcal{A}$ finitely determined map-germs $f_t: (\mathbb C^2,0) \to (\mathbb C^3,0),$ in terms of the constancy of only one invariant, the Milnor number of the double point locus.